PARTICLE FILTERING - Adaptive Signal Processing: Next Generation Solutions - page 309

of this type as density-assisted PFs (DAPFs) [24], which are a generalization of the
Gaussian and Gaussian sum PFs. The main advantages of DAPFs is that they do
not necessarily use resampling in the sense carried out by standard PFs and they do
not share the limitation regarding the estimation of constant model parameters. Here
we explain how we can use DAPFs when we have constant parameters in the model.
Let x ( n 1) ¼ {x

( m ) ( n 1), u ( m ) ( n 1), w ( m ) ( n 1)} m¼ 1 be the randommeasure

( m ) ( n 2 1) the particles of x( n 2 1) and u ,
respectively, and w ( m ) ( n 2 1) their associated weights. If we denote the approximat-
ing density of f (x( n 2 1), uj y(1 : n 2 1)) by p ( f ( n 2 1)), where f ( n 2 1) are the
parameters of the approximating density, the steps of the density assisted particle fil-
tering are the following.

( m ) ( n 2 1) and u

at time instant n 2 1, x

1. Draw particles according to

( m ) ( n 1), u ( m ) ( n 1)} p ( f ( n 1)) :

{x

2. Draw particles according to

( m ) ( n ) f (x( n ) j x ( m ) ( n 1), u ( m ) ( n 1)) :

x

( m ) ( n 1) :
4. Update and normalize the weights

( m ) ( n ) ¼ u

3. Set u

w ( m ) ( n ) / f ( y( n ) j x

( m ) ( n ), u ( m ) ( n )) :

5. Estimate the parameters, f ( n ), of the density from

( m ) ( n ), w ( m ) ( n )} m¼ 1 : ( m ) ( n ), u

x ( n ) ¼ {x

The problem of standard PFs regarding constant parameters is avoided in the first
step by drawing particles of the constants from an approximation of the posterior. It is
important to note that one can combine standard PFs and DAPFs, in that we apply the
standard PFs for the dynamic variables and DAPFs for the constant parameters. We
show this in the next example.

B EXAMPLE 5.10
Again we have the problem from Example 5.9, where we track a target in a two-
dimensional plane. We assume that the marginal posterior density of the bias
vector u is a Gaussian density. Suppose that at time instant n 2 1, we have the
random measure x ( n 1) ¼ {x

( m ) ( n 1), w ( m ) ( n 1)} m¼ 1 .From
the random measure, we can compute the parameters of the approximating

( m ) ( n 1), u